# Properties

 Label 1089.6.a.p Level $1089$ Weight $6$ Character orbit 1089.a Self dual yes Analytic conductor $174.658$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$174.657979776$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 7) q^{2} + ( - 13 \beta + 25) q^{4} + ( - 10 \beta - 24) q^{5} + ( - 62 \beta - 42) q^{7} + ( - 71 \beta + 55) q^{8}+O(q^{10})$$ q + (-b + 7) * q^2 + (-13*b + 25) * q^4 + (-10*b - 24) * q^5 + (-62*b - 42) * q^7 + (-71*b + 55) * q^8 $$q + ( - \beta + 7) q^{2} + ( - 13 \beta + 25) q^{4} + ( - 10 \beta - 24) q^{5} + ( - 62 \beta - 42) q^{7} + ( - 71 \beta + 55) q^{8} + ( - 36 \beta - 88) q^{10} + (74 \beta + 28) q^{13} + ( - 330 \beta + 202) q^{14} + ( - 65 \beta + 153) q^{16} + (372 \beta - 550) q^{17} + (852 \beta - 12) q^{19} + (192 \beta + 440) q^{20} + (330 \beta - 46) q^{23} + (580 \beta - 1749) q^{25} + (416 \beta - 396) q^{26} + ( - 198 \beta + 5398) q^{28} + ( - 1492 \beta + 1094) q^{29} + (1600 \beta - 6040) q^{31} + (1729 \beta - 169) q^{32} + (2782 \beta - 6826) q^{34} + (2528 \beta + 5968) q^{35} + ( - 2816 \beta + 454) q^{37} + (5124 \beta - 6900) q^{38} + (1864 \beta + 4360) q^{40} + ( - 8 \beta + 18246) q^{41} + (3112 \beta - 6440) q^{43} + (2026 \beta - 2962) q^{46} + (390 \beta - 22066) q^{47} + (9052 \beta + 15709) q^{49} + (5229 \beta - 16883) q^{50} + (524 \beta - 6996) q^{52} + (7102 \beta + 2536) q^{53} + (3974 \beta + 32906) q^{56} + ( - 10046 \beta + 19594) q^{58} + ( - 1980 \beta + 2384) q^{59} + ( - 2026 \beta + 13664) q^{61} + (15640 \beta - 55080) q^{62} + (12623 \beta - 19911) q^{64} + ( - 2796 \beta - 6592) q^{65} + ( - 12704 \beta - 13908) q^{67} + (11614 \beta - 52438) q^{68} + (9200 \beta + 21552) q^{70} + (4354 \beta - 17870) q^{71} + ( - 5568 \beta + 26174) q^{73} + ( - 17350 \beta + 25706) q^{74} + (10380 \beta - 88908) q^{76} + ( - 11426 \beta + 14138) q^{79} + (680 \beta + 1528) q^{80} + ( - 18294 \beta + 127786) q^{82} + (21960 \beta + 28740) q^{83} + ( - 7148 \beta - 16560) q^{85} + (25112 \beta - 69976) q^{86} + ( - 26704 \beta + 40454) q^{89} + ( - 9432 \beta - 37880) q^{91} + (4558 \beta - 35470) q^{92} + (24406 \beta - 157582) q^{94} + ( - 28848 \beta - 67872) q^{95} + (9924 \beta - 125746) q^{97} + (38603 \beta + 37547) q^{98}+O(q^{100})$$ q + (-b + 7) * q^2 + (-13*b + 25) * q^4 + (-10*b - 24) * q^5 + (-62*b - 42) * q^7 + (-71*b + 55) * q^8 + (-36*b - 88) * q^10 + (74*b + 28) * q^13 + (-330*b + 202) * q^14 + (-65*b + 153) * q^16 + (372*b - 550) * q^17 + (852*b - 12) * q^19 + (192*b + 440) * q^20 + (330*b - 46) * q^23 + (580*b - 1749) * q^25 + (416*b - 396) * q^26 + (-198*b + 5398) * q^28 + (-1492*b + 1094) * q^29 + (1600*b - 6040) * q^31 + (1729*b - 169) * q^32 + (2782*b - 6826) * q^34 + (2528*b + 5968) * q^35 + (-2816*b + 454) * q^37 + (5124*b - 6900) * q^38 + (1864*b + 4360) * q^40 + (-8*b + 18246) * q^41 + (3112*b - 6440) * q^43 + (2026*b - 2962) * q^46 + (390*b - 22066) * q^47 + (9052*b + 15709) * q^49 + (5229*b - 16883) * q^50 + (524*b - 6996) * q^52 + (7102*b + 2536) * q^53 + (3974*b + 32906) * q^56 + (-10046*b + 19594) * q^58 + (-1980*b + 2384) * q^59 + (-2026*b + 13664) * q^61 + (15640*b - 55080) * q^62 + (12623*b - 19911) * q^64 + (-2796*b - 6592) * q^65 + (-12704*b - 13908) * q^67 + (11614*b - 52438) * q^68 + (9200*b + 21552) * q^70 + (4354*b - 17870) * q^71 + (-5568*b + 26174) * q^73 + (-17350*b + 25706) * q^74 + (10380*b - 88908) * q^76 + (-11426*b + 14138) * q^79 + (680*b + 1528) * q^80 + (-18294*b + 127786) * q^82 + (21960*b + 28740) * q^83 + (-7148*b - 16560) * q^85 + (25112*b - 69976) * q^86 + (-26704*b + 40454) * q^89 + (-9432*b - 37880) * q^91 + (4558*b - 35470) * q^92 + (24406*b - 157582) * q^94 + (-28848*b - 67872) * q^95 + (9924*b - 125746) * q^97 + (38603*b + 37547) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10})$$ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 $$2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100})$$ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 - 212 * q^10 + 130 * q^13 + 74 * q^14 + 241 * q^16 - 728 * q^17 + 828 * q^19 + 1072 * q^20 + 238 * q^23 - 2918 * q^25 - 376 * q^26 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 + 1391 * q^32 - 10870 * q^34 + 14464 * q^35 - 1908 * q^37 - 8676 * q^38 + 10584 * q^40 + 36484 * q^41 - 9768 * q^43 - 3898 * q^46 - 43742 * q^47 + 40470 * q^49 - 28537 * q^50 - 13468 * q^52 + 12174 * q^53 + 69786 * q^56 + 29142 * q^58 + 2788 * q^59 + 25302 * q^61 - 94520 * q^62 - 27199 * q^64 - 15980 * q^65 - 40520 * q^67 - 93262 * q^68 + 52304 * q^70 - 31386 * q^71 + 46780 * q^73 + 34062 * q^74 - 167436 * q^76 + 16850 * q^79 + 3736 * q^80 + 237278 * q^82 + 79440 * q^83 - 40268 * q^85 - 114840 * q^86 + 54204 * q^89 - 85192 * q^91 - 66382 * q^92 - 290758 * q^94 - 164592 * q^95 - 241568 * q^97 + 113697 * q^98

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.37228 −2.37228
3.62772 0 −18.8397 −57.7228 0 −251.081 −184.432 0 −209.402
1.2 9.37228 0 55.8397 −0.277187 0 105.081 223.432 0 −2.59787
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.6.a.p 2
3.b odd 2 1 363.6.a.f 2
11.b odd 2 1 99.6.a.d 2
33.d even 2 1 33.6.a.e 2
132.d odd 2 1 528.6.a.o 2
165.d even 2 1 825.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 33.d even 2 1
99.6.a.d 2 11.b odd 2 1
363.6.a.f 2 3.b odd 2 1
528.6.a.o 2 132.d odd 2 1
825.6.a.c 2 165.d even 2 1
1089.6.a.p 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - 13T_{2} + 34$$ T2^2 - 13*T2 + 34 $$T_{5}^{2} + 58T_{5} + 16$$ T5^2 + 58*T5 + 16 $$T_{7}^{2} + 146T_{7} - 26384$$ T7^2 + 146*T7 - 26384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 13T + 34$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 58T + 16$$
$7$ $$T^{2} + 146T - 26384$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 130T - 40952$$
$17$ $$T^{2} + 728 T - 1009172$$
$19$ $$T^{2} - 828 T - 5817312$$
$23$ $$T^{2} - 238T - 884264$$
$29$ $$T^{2} - 696 T - 18243924$$
$31$ $$T^{2} + 10480 T + 6337600$$
$37$ $$T^{2} + 1908 T - 64511196$$
$41$ $$T^{2} - 36484 T + 332770036$$
$43$ $$T^{2} + 9768 T - 56044032$$
$47$ $$T^{2} + 43742 T + 477085816$$
$53$ $$T^{2} - 12174 T - 379065264$$
$59$ $$T^{2} - 2788 T - 30400064$$
$61$ $$T^{2} - 25302 T + 126184224$$
$67$ $$T^{2} + 40520 T - 921013232$$
$71$ $$T^{2} + 31386 T + 89872392$$
$73$ $$T^{2} - 46780 T + 291320452$$
$79$ $$T^{2} - 16850 T - 1006085552$$
$83$ $$T^{2} - 79440 T - 2400814800$$
$89$ $$T^{2} - 54204 T - 5148586428$$
$97$ $$T^{2} + 241568 T + 13776267004$$